Subgroups — #13098522

Order 1600.

759 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₂₅	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂²⋊Dic₂₅	C₂	1	6	yes	maximal, complemented
3	C₂⁴×C₂₅	—	2	12	no	—
4	C₂⁴×C₂₅	C₄	1	4	yes	complemented
5	C₂⁴×C₂₅	C₂²	1	3	yes	—
6	C₂²×Dic₂₅	C₂²	1	4	yes	complemented
7	C₂²⋊Dic₂₅	—	2	12	no	—
8	#65536	—	5	1	no	maximal
9	C₂³×C₂₅	—	2	32	no	—
10	C₂³×C₂₅	—	2	24	no	—
11	C₂³×C₂₅	—	2	12	no	—
12	C₂³×C₂₅	C₂×C₄	1	6	yes	—
13	C₂³×C₂₅	D₄	1	12	yes	—
14	C₂³×C₂₅	C₂³	1	1	yes	characteristic
15	C₂×Dic₂₅	—	2	12	no	—
16	C₂⁵×C₅	D₅	1	1	yes	characteristic
17	C₂×C₂²⋊Dic₅	—	5	6	no	—
18	C₂²×C₂₅	—	2	32	no	—
19	C₂²×C₂₅	—	2	12	no	—
20	C₂²×C₂₅	—	2	24	no	—
21	C₂²×C₂₅	C₂²⋊C₄	1	12	yes	complemented
22	C₂²×C₂₅	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₂₅	C₂×D₄	1	6	yes	—
24	Dic₂₅	—	4	4	no	—
25	C₂⁴×C₅	—	2	12	no	—
26	C₂⁴×C₅	Dic₅	1	4	yes	—
27	C₂⁴×C₅	C₂×D₅	1	3	yes	—
28	C₂²×Dic₅	—	5	4	no	—
29	C₂²⋊Dic₅	—	10	12	no	—
30	#379	—	25	1	no	Sylow
31	C₂×C₂₅	—	2	12	no	—
32	C₂×C₂₅	C₂×C₂²⋊C₄	1	3	yes	—
33	C₂×C₂₅	#120	1	4	yes	—
34	C₂³×C₅	—	2	32	no	—
35	C₂³×C₅	—	2	24	no	—
36	C₂³×C₅	—	2	12	no	—
37	C₂³×C₅	C₂×Dic₅	1	6	yes	—
38	C₂³×C₅	C₂²⋊D₅	1	12	yes	—
39	C₂³×C₅	C₂²×D₅	1	1	yes	characteristic
40	C₂×Dic₅	—	10	12	no	—
41	C₂⁵	D₂₅	1	1	yes	characteristic
42	C₂×C₂²⋊C₄	—	25	6	no	—
43	C₂₅	#379	1	1	yes	characteristic, complemented, Sylow
44	C₂²×C₅	—	2	32	no	—
45	C₂²×C₅	—	2	12	no	—
46	C₂²×C₅	—	2	24	no	—
47	C₂²×C₅	C₂²⋊Dic₅	1	12	yes	—
48	C₂²×C₅	C₂²×Dic₅	1	1	yes	characteristic
49	C₂²×C₅	C₂×C₂²⋊D₅	1	6	yes	—
50	Dic₅	—	20	4	no	—
51	C₂⁴	—	2	12	no	—
52	C₂⁴	Dic₂₅	1	4	yes	complemented
53	C₂⁴	C₂×D₂₅	1	3	yes	—
54	C₂²×C₄	—	25	4	no	—
55	C₂²⋊C₄	—	50	12	no	—
56	C₂×C₅	—	2	12	no	—
57	C₂×C₅	C₂×C₂²⋊Dic₅	1	3	yes	—
58	C₂×C₅	#4099	1	4	yes	—
59	C₂³	—	2	32	no	—
60	C₂³	—	2	24	no	—
61	C₂³	—	2	12	no	—
62	C₂³	C₂×Dic₂₅	1	6	yes	—
63	C₂³	C₂²⋊D₂₅	1	12	yes	—
64	C₂³	C₂²×D₂₅	1	1	yes	characteristic
65	C₂×C₄	—	50	12	no	—
66	C₅	#65536	1	1	yes	characteristic
67	C₂²	—	2	32	no	—
68	C₂²	—	2	12	no	—
69	C₂²	—	2	24	no	—
70	C₂²	C₂²⋊Dic₂₅	1	12	yes	complemented
71	C₂²	C₂²×Dic₂₅	1	1	yes	characteristic
72	C₂²	C₂×C₂²⋊D₂₅	1	6	yes	—
73	C₄	—	100	4	no	—
74	C₂	—	2	12	no	—
75	C₂	C₂×C₂²⋊Dic₂₅	1	3	yes	—
76	C₂	#11716515	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₂₅
commutator	C₂²×C₂₅	C₂²×C₄
Frattini, socle	C₂³×C₅	C₂²×D₅
Fitting	C₂⁵×C₂₅	C₂
prime radical	C₂⁵×C₅	D₅
cosocle	C₂³×C₂₅	C₂³
hypercenter	C₂⁵	D₂₅
hypocenter	C₂₅	#379
LCS	G ▷ C₂²×C₂₅ ▷ C₂₅
UCS	C₁ ◁ C₂³ ◁ C₂⁵

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. C₂×C₂²⋊Dic₂₅	C₂	semidirect
4. C₂⁴×C₂₅	C₄	semidirect
6. C₂²×Dic₂₅	C₂²	semidirect
21. C₂²×C₂₅	C₂²⋊C₄	semidirect
43. C₂₅	#379	semidirect
52. C₂⁴	Dic₂₅	semidirect
70. C₂²	C₂²⋊Dic₂₅	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂⁵×C₂₅	2
C₂×C₂²⋊Dic₂₅	2
#65536	5